Physical Chemistry week 1 Monday January 7th, 2013 page 1
Chapter 7: one component phase equilibrium
β
μi =μδu condition for phase equilibrium
Assume a system has a saturated solution of NaCl and a solid phase of pure NaCl, with T and P constant.
Once the solution is saturated, the quantity of NaCl(s) is irrelevant.
χ is the mole fraction, Greek letter lowercase chi (pronounced kie)
χαj
α
χj = α
χtot
so χαj is intensive
f is the number of degrees of freedom, formerly called variance
f is the number of independent intensive variables needed to specify the state of the system
independent intensive variables: T, P, χj (χ1 + χ2=1)
c
∑ χi =1
c is the number of components
i=1
total number of intensive variables = 1 (for T) + 1 (for P) + p(c-1) (for composition variables)
P is pressure and p is the number of phases
f = 2 + p(c-1)
1. Assume every component (chemical species) is present in every phase.
2. Assume no chemical reaction occurs among the phases or components
β
At equilibrium for components j, we have: μαj = μj = μδj
There are p-1 equations to satisfy (independent equations)
If A=B and B=C then A=C so stating A=C is redundant.
total number of independent equations = c(p-1)
f=(2+p(c-1))
total number of intensive variables
-c(p-1) number of equations(relations) among intensive variables
f = 2 + pc – p – cp + c = c – p + 2
phase rule
very important equation
For a pure component (like H2O(l)): c=1 p=1 f=2 (T and P)
Note: The phase rule holds even when every species does not appear in every phase.
f=(2+p(c-1) -1 (for missing species) –(c(p-1) -1 (for missing species)
f = 2 + pc + p -1 -1 –cp +p +1 (the extra -1 and +1 cancel out)
f=2-p+c = c-p+2
saturated aqueous sucrose solution: p=2 c=2
The phase rule in systems with reactions
Let r = the number of independent chemical reactions
Independent chemical reactions can’t be written as a sum of the other chemical reactions, like we do
with Hess’s Law.
f= c – p + 2 – r – a
a is the number of additional restrictions, like electroneutrality
cind is the number of independent components.
cind = c – r – a
f = cind – p + 2
Example: For pure liquid water c = 1 p = 1 f = 1 – 1 + 2 = 2
water dissociates
H2O ⇌ H+ + OH-
r = 1 c = 3 p = 1 a = 1(electroneutrality)
χH+ = χOH- to maintain electroneutrality
a is usually 1 or 0, unless there’s an electric force field
cind = 3 – 1 – 1 = 1
f = cind – p + 2 = 1 – 1 + 2 = 2
phase diagram for pure H2O: Graph P in atm on the vertical axis and T in ℃. A curve goes from the
origin to a point, the triple point. A curve goes up and left from the triple point. Left of these two
curves is where water is solid. The degrees of freedom within the solid phase is 2. A curve goes from
the triple point up and right and ends at a point called the critical point. Below this curve and the first
curve is the vapor phase. The degrees of freedom within the vapor phase is 2. The rest of the graph is
the liquid phase. The degrees of freedom within the liquid phase is 2. At any point such as the triple
point or critical point, the degrees of freedom is 0. Along any curve the degrees of freedom is 1.
Where f=0 , you can’t change T or P, or you won’t be at the same point any more.
Where f=1, you can change T or P, but then the other variables has to be a certain value to stay on the
curve.
Where f=2, you can change T or P or both and still be in the same phase.
Tc is the temperature at the critical point.
Pc is the pressure at the critical point.
For H2O Pc = 218 atm
Tc = 374℃